Require Import
        HoTT.Basics
        HoTT.DProp
        HoTT.BoundedSearch
        HoTT.Types.Universe
        HoTT.Types.Sum
        HoTT.Types.Arrow
        HoTT.Spaces.Finite
        HoTT.ExcludedMiddle.

Require Import
        HoTT.Classes.interfaces.abstract_algebra
        HoTT.Classes.interfaces.orders
        HoTT.Classes.interfaces.integers
        HoTT.Classes.interfaces.rationals
        HoTT.Classes.interfaces.naturals
        HoTT.Classes.interfaces.cauchy
        HoTT.Classes.interfaces.archimedean
        HoTT.Classes.interfaces.round
        HoTT.Classes.interfaces.naturals
        HoTT.Classes.implementations.peano_naturals
        HoTT.Classes.orders.orders
        HoTT.Classes.orders.archimedean
        HoTT.Classes.orders.rings
        HoTT.Classes.orders.dec_fields
        HoTT.Classes.orders.semirings
        HoTT.Classes.orders.lattices
        HoTT.Classes.theory.rings
        HoTT.Classes.theory.apartness
        HoTT.Classes.theory.rationals.

(* Strangely, it seems that combining the next import with the above list breaks some instance search? *)
Require Import
        HoTT.Classes.orders.fields
        HoTT.Classes.theory.fields
        HoTT.Classes.theory.dec_fields.

Local Open Scope type_scope.

Section locator.
  Context (Q : Type).
  Context `{Qrats : Rationals Q}.
  Context {Qdec_paths : DecidablePaths Q}.
  Context {Qtriv : TrivialApart Q}.
  Context `{!Trichotomy (<)}.
  Context (F : Type).
  Context `{Forderedfield : OrderedField F}.
  Context {Fabs : Abs F}.
  Context {Farchimedean : ArchimedeanProperty Q F}.
  Context {Fcomplete : IsComplete Q F}.
  Context {Qroundup : RoundUpStrict Q}.

  Context `{Funext} `{Univalence}.

  (* Assume we have enumerations of the rationals, and of pairs of ordered rationals. *)
  Context (Q_eq : nat <~> Q).
  Context (QQpos_eq : nat <~> Q * Qpos Q).

  Instance qinc : Cast Q F := rationals_to_field Q F.
  (* TODO The following two instances should probably come from the `Rationals` instance. *)
  Context (cast_pres_ordering : StrictlyOrderPreserving qinc)
          (qinc_strong_presving : IsSemiRingStrongPreserving qinc).
  Existing Instance cast_pres_ordering.
  Existing Instance qinc_strong_presving.

  (* Definition of a locator for a fixed real number. *)
  Definition locator (x : F) := forall q r : Q, q < r -> (' q < x) + (x < ' r).

  (* Alternative definition; see equivalence below *)
  Record locator' (x : F) :=
    { locates_right : forall q r : Q, q < r -> DHProp
    ; locates_right_true : forall q r : Q, forall nu : q < r, locates_right q r nu -> ' q < x
    ; locates_right_false : forall q r : Q, forall nu : q < r, ~ locates_right q r nu -> x < ' r
    }.

  Arguments locates_right       [x] _ [q] [r] _.
  Arguments locates_right_true  [x] _ [q] [r] _.
  Arguments locates_right_false [x] _ [q] [r] _.

  Definition locates_left {x : F} (l : locator' x) {q r : Q} : q < r -> DHProp :=
    fun nu => Build_DHProp (Build_HProp (~ (locates_right l nu))) _.

  Section classical.
    Context `{ExcludedMiddle}.

    Lemma all_reals_locators (x : F) : locator x.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
locator x
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
locator x
      intros q r ltqr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
q, r:Q
ltqr:q < r
(' q < x) + (x < ' r)
      case (LEM (' q < x)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
q, r:Q
ltqr:q < r
IsHProp (' q < x)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
q, r:Q
ltqr:q < r
' q < x -> (' q < x) + (x < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
q, r:Q
ltqr:q < r
~ (' q < x) -> (' q < x) + (x < ' r)
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
q, r:Q
ltqr:q < r
IsHProp (' q < x) apply _.
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
q, r:Q
ltqr:q < r
' q < x -> (' q < x) + (x < ' r) exact inl.
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
q, r:Q
ltqr:q < r
~ (' q < x) -> (' q < x) + (x < ' r) intros notlt.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
q, r:Q
ltqr:q < r
notlt:~ (' q < x)
(' q < x) + (x < ' r)
        apply inr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
q, r:Q
ltqr:q < r
notlt:~ (' q < x)
x < ' r
        assert (ltqr' : ' q < ' r)
          by auto.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
q, r:Q
ltqr:q < r
notlt:~ (' q < x)
ltqr':' q < ' r
x < ' r
        exact (nlt_lt_trans notlt ltqr').
    Qed.

  End classical.

  Section rational.
    Context (s : Q).

    Lemma locator_left : locator (' s).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s:Q
locator (' s)
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s:Q
locator (' s)
      intros q r ltqr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
(' q < ' s) + (' s < ' r)
      destruct (trichotomy _ q s) as [ltqs|[eqqs|ltsq]].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
ltqs:q < s
(' q < ' s) + (' s < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
eqqs:q = s
(' q < ' s) + (' s < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
ltsq:s < q
(' q < ' s) + (' s < ' r)
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
ltqs:q < s
(' q < ' s) + (' s < ' r) apply inl.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
ltqs:q < s
' q < ' s apply (strictly_order_preserving _); assumption.
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
eqqs:q = s
(' q < ' s) + (' s < ' r) rewrite eqqs in ltqr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:s < r
eqqs:q = s
(' q < ' s) + (' s < ' r) apply inr, (strictly_order_preserving _); assumption.
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
ltsq:s < q
(' q < ' s) + (' s < ' r) apply inr, (strictly_order_preserving _), (transitivity ltsq ltqr); assumption.
    Qed.

    Definition locator_second : locator (' s).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s:Q
locator (' s)
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s:Q
locator (' s)
      intros q r ltqr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
(' q < ' s) + (' s < ' r)
      destruct (trichotomy _ s r) as [ltsr|[eqsr|ltrs]].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
ltsr:s < r
(' q < ' s) + (' s < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
eqsr:s = r
(' q < ' s) + (' s < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
ltrs:r < s
(' q < ' s) + (' s < ' r)
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
ltsr:s < r
(' q < ' s) + (' s < ' r) apply inr, (strictly_order_preserving _); assumption.
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
eqsr:s = r
(' q < ' s) + (' s < ' r) rewrite <- eqsr in ltqr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < s
eqsr:s = r
(' q < ' s) + (' s < ' r) apply inl, (strictly_order_preserving _); assumption.
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
ltrs:r < s
(' q < ' s) + (' s < ' r) apply inl, (strictly_order_preserving _), (transitivity ltqr ltrs).
    Qed.

  End rational.

  Section logic.

    Context {x : F}.

    Definition locator_locator' : locator x -> locator' x.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
locator x -> locator' x
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
locator x -> locator' x
      intros l.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
locator' x
      refine (Build_locator' x (fun q r nu => Build_DHProp (Build_HProp (is_inl (l q r nu))) _) _ _).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
forall (q r : Q) (nu : q < r),
{|
  dhprop_hprop := Build_HProp (is_inl (l q r nu));
  dec_dhprop := decidable_is_inl (l q r nu)
|} -> ' q < x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
forall (q r : Q) (nu : q < r),
~
{|
  dhprop_hprop := Build_HProp (is_inl (l q r nu));
  dec_dhprop := decidable_is_inl (l q r nu)
|} -> x < ' r
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
forall (q r : Q) (nu : q < r),
{|
  dhprop_hprop := Build_HProp (is_inl (l q r nu));
  dec_dhprop := decidable_is_inl (l q r nu)
|} -> ' q < x intros q r nu.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r
{|
  dhprop_hprop := Build_HProp (is_inl (l q r nu));
  dec_dhprop := decidable_is_inl (l q r nu)
|} -> ' q < x simpl.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r
is_inl (l q r nu) -> ' q < x apply un_inl.
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
forall (q r : Q) (nu : q < r),
~
{|
  dhprop_hprop := Build_HProp (is_inl (l q r nu));
  dec_dhprop := decidable_is_inl (l q r nu)
|} -> x < ' r intros q r nu.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r
~
{|
  dhprop_hprop := Build_HProp (is_inl (l q r nu));
  dec_dhprop := decidable_is_inl (l q r nu)
|} -> x < ' r simpl.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r
~ is_inl (l q r nu) -> x < ' r destruct (l q r nu) as [ltqx|].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r
ltqx:' q < x
~ is_inl (inl ltqx) -> x < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r
l0:x < ' r
~ is_inl (inr l0) -> x < ' r
        +
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r
ltqx:' q < x
~ is_inl (inl ltqx) -> x < ' r simpl; intros f; destruct (f tt).
        +
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r
l0:x < ' r
~ is_inl (inr l0) -> x < ' r intros ?; assumption.
    Defined.

    Definition locator'_locator : locator' x -> locator x.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
locator' x -> locator x
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
locator' x -> locator x
      intros l' q r nu.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
q, r:Q
nu:q < r
(' q < x) + (x < ' r)
      destruct (dec (locates_right l' nu)) as [yes|no].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
q, r:Q
nu:q < r
yes:locates_right l' nu
(' q < x) + (x < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
q, r:Q
nu:q < r
no:~ locates_right l' nu
(' q < x) + (x < ' r)
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
q, r:Q
nu:q < r
yes:locates_right l' nu
(' q < x) + (x < ' r) apply inl.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
q, r:Q
nu:q < r
yes:locates_right l' nu
' q < x exact (locates_right_true l' nu yes).
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
q, r:Q
nu:q < r
no:~ locates_right l' nu
(' q < x) + (x < ' r) apply inr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
q, r:Q
nu:q < r
no:~ locates_right l' nu
x < ' r exact (locates_right_false l' nu no).
    Defined.
  End logic.

  Section logic2.

    Context {x : F}.

    Coercion locator_locator' : locator >-> locator'.

    Definition locator_locator'_locator (l : locator x) : locator'_locator (locator_locator' l) = l.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
locator'_locator l = l
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
locator'_locator l = l
      apply path_forall; intros q.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q:Q
locator'_locator l q = l q
      apply path_forall; intros r.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
locator'_locator l q r = l q r
      apply path_forall; intros nu.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r
locator'_locator l q r nu = l q r nu
      unfold locator'_locator, locator_locator'.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r
match
  dec
    (locates_right
       {|
         locates_right :=
           fun (q r : Q) (nu : q < r) =>
           {|
             dhprop_hprop :=
               Build_HProp (is_inl (l q r nu));
             dec_dhprop := decidable_is_inl (l q r nu)
           |};
         locates_right_true :=
           fun (q r : Q) (nu : q < r) =>
           un_inl (l q r nu);
         locates_right_false :=
           fun (q r : Q) (nu : q < r) =>
           match
             l q r nu as s
             return (~ is_inl s -> x < ' r)
           with
           | inl _ =>
               fun f : ~ Unit =>
               match f tt return (x < ' r) with
               end
           | inr b => fun _ : ~ is_inl (inr b) => b
           end
       |} nu)
with
| inl a =>
    inl
      (locates_right_true
         {|
           locates_right :=
             fun (q r : Q) (nu : q < r) =>
             {|
               dhprop_hprop :=
                 Build_HProp (is_inl (l q r nu));
               dec_dhprop :=
                 decidable_is_inl (l q r nu)
             |};
           locates_right_true :=
             fun (q r : Q) (nu : q < r) =>
             un_inl (l q r nu);
           locates_right_false :=
             fun (q r : Q) (nu : q < r) =>
             match
               l q r nu as s
               return (~ is_inl s -> x < ' r)
             with
             | inl _ =>
                 fun f : ~ Unit =>
                 match f tt return (x < ' r) with
                 end
             | inr b => fun _ : ~ is_inl (inr b) => b
             end
         |} nu a)
| inr b =>
    inr
      (locates_right_false
         {|
           locates_right :=
             fun (q r : Q) (nu : q < r) =>
             {|
               dhprop_hprop :=
                 Build_HProp (is_inl (l q r nu));
               dec_dhprop :=
                 decidable_is_inl (l q r nu)
             |};
           locates_right_true :=
             fun (q r : Q) (nu : q < r) =>
             un_inl (l q r nu);
           locates_right_false :=
             fun (q r : Q) (nu : q < r) =>
             match
               l q r nu as s
               return (~ is_inl s -> x < ' r)
             with
             | inl _ =>
                 fun f : ~ Unit =>
                 match f tt return (x < ' r) with
                 end
             | inr b0 =>
                 fun _ : ~ is_inl (inr b0) => b0
             end
         |} nu b)
end = l q r nu simpl.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r
match dec (is_inl (l q r nu)) with
| inl a => inl (un_inl (l q r nu) a)
| inr b =>
    inr
      (match
         l q r nu as s return (~ is_inl s -> x < ' r)
       with
       | inl _ =>
           fun f : ~ Unit =>
           match f tt return (x < ' r) with
           end
       | inr b0 => fun _ : ~ Empty => b0
       end b)
end = l q r nu
      destruct (l q r nu); auto.
    Qed.

    Definition locsig : _ <~> locator' x := ltac:(issig).

    Lemma locator'_locator_locator' (l' : locator' x)
      : locator_locator' (locator'_locator l') = l'.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
locator'_locator l' = l'
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
locator'_locator l' = l'
      enough (p : locsig ^-1 (locator_locator' (locator'_locator l')) = locsig ^-1 l').
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
p:locsig^-1 (locator'_locator l') = locsig^-1 l'
locator'_locator l' = l'
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
locsig^-1 (locator'_locator l') = locsig^-1 l'
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
p:locsig^-1 (locator'_locator l') = locsig^-1 l'
locator'_locator l' = l' refine (equiv_inj (locsig ^-1) p).
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
locsig^-1 (locator'_locator l') = locsig^-1 l' unfold locsig; simpl.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
(fun (q r : Q) (nu : q < r) =>
 {|
   dhprop_hprop :=
     Build_HProp (is_inl (locator'_locator l' q r nu));
   dec_dhprop :=
     decidable_is_inl (locator'_locator l' q r nu)
 |};
fun (q r : Q) (nu : q < r) =>
un_inl (locator'_locator l' q r nu);
fun (q r : Q) (nu : q < r) =>
match
  locator'_locator l' q r nu as s
  return (~ is_inl s -> x < ' r)
with
| inl _ =>
    fun f : ~ Unit =>
    match f tt return (x < ' r) with
    end
| inr b => fun _ : ~ Empty => b
end) =
(locates_right l'; locates_right_true l';
locates_right_false l')
        destruct l'; unfold locator'_locator, locator_locator'; simpl.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
locates_right0:forall q r : Q, q < r -> DHProp
locates_right_true0:forall (q r : Q) (nu : q < r),
locates_right0 q r nu -> ' q < x
locates_right_false0:forall (q r : Q) (nu : q < r),
~ locates_right0 q r nu ->
x < ' r
(fun (q r : Q) (nu : q < r) =>
 {|
   dhprop_hprop :=
     Build_HProp
       (is_inl
          match dec (locates_right0 q r nu) with
          | inl a =>
              inl (locates_right_true0 q r nu a)
          | inr b =>
              inr (locates_right_false0 q r nu b)
          end);
   dec_dhprop :=
     decidable_is_inl
       match dec (locates_right0 q r nu) with
       | inl a => inl (locates_right_true0 q r nu a)
       | inr b => inr (locates_right_false0 q r nu b)
       end
 |};
fun (q r : Q) (nu : q < r) =>
un_inl
  match dec (locates_right0 q r nu) with
  | inl a => inl (locates_right_true0 q r nu a)
  | inr b => inr (locates_right_false0 q r nu b)
  end;
fun (q r : Q) (nu : q < r) =>
match
  match dec (locates_right0 q r nu) with
  | inl a => inl (locates_right_true0 q r nu a)
  | inr b => inr (locates_right_false0 q r nu b)
  end as s return (~ is_inl s -> x < ' r)
with
| inl _ =>
    fun f : ~ Unit =>
    match f tt return (x < ' r) with
    end
| inr b => fun _ : ~ Empty => b
end) =
(locates_right0; locates_right_true0;
locates_right_false0)
        apply path_sigma_hprop; simpl.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
locates_right0:forall q r : Q, q < r -> DHProp
locates_right_true0:forall (q r : Q) (nu : q < r),
locates_right0 q r nu -> ' q < x
locates_right_false0:forall (q r : Q) (nu : q < r),
~ locates_right0 q r nu ->
x < ' r
(fun (q r : Q) (nu : q < r) =>
 {|
   dhprop_hprop :=
     Build_HProp
       (is_inl
          match dec (locates_right0 q r nu) with
          | inl a =>
              inl (locates_right_true0 q r nu a)
          | inr b =>
              inr (locates_right_false0 q r nu b)
          end);
   dec_dhprop :=
     decidable_is_inl
       match dec (locates_right0 q r nu) with
       | inl a => inl (locates_right_true0 q r nu a)
       | inr b => inr (locates_right_false0 q r nu b)
       end
 |}) = locates_right0
        apply path_forall; intro q;
          apply path_forall; intro r;
            apply path_arrow; intro nu.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
locates_right0:forall q r : Q, q < r -> DHProp
locates_right_true0:forall (q r : Q) (nu : q < r),
locates_right0 q r nu -> ' q < x
locates_right_false0:forall (q r : Q) (nu : q < r),
~ locates_right0 q r nu ->
x < ' r
q, r:Q
nu:q < r
{|
  dhprop_hprop :=
    Build_HProp
      (is_inl
         match dec (locates_right0 q r nu) with
         | inl a => inl (locates_right_true0 q r nu a)
         | inr b =>
             inr (locates_right_false0 q r nu b)
         end);
  dec_dhprop :=
    decidable_is_inl
      match dec (locates_right0 q r nu) with
      | inl a => inl (locates_right_true0 q r nu a)
      | inr b => inr (locates_right_false0 q r nu b)
      end
|} = locates_right0 q r nu
        apply equiv_path_dhprop; simpl.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
locates_right0:forall q r : Q, q < r -> DHProp
locates_right_true0:forall (q r : Q) (nu : q < r),
locates_right0 q r nu -> ' q < x
locates_right_false0:forall (q r : Q) (nu : q < r),
~ locates_right0 q r nu ->
x < ' r
q, r:Q
nu:q < r
is_inl
  match dec (locates_right0 q r nu) with
  | inl a => inl (locates_right_true0 q r nu a)
  | inr b => inr (locates_right_false0 q r nu b)
  end = locates_right0 q r nu
        rewrite (path_dec (locates_right0 q r nu)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
locates_right0:forall q r : Q, q < r -> DHProp
locates_right_true0:forall (q r : Q) (nu : q < r),
locates_right0 q r nu -> ' q < x
locates_right_false0:forall (q r : Q) (nu : q < r),
~ locates_right0 q r nu ->
x < ' r
q, r:Q
nu:q < r
is_inl
  match dec (locates_right0 q r nu) with
  | inl a => inl (locates_right_true0 q r nu a)
  | inr b => inr (locates_right_false0 q r nu b)
  end = is_inl (dec (locates_right0 q r nu))
        destruct (dec (locates_right0 q r nu)); auto.
    Qed.

    Definition equiv_locator_locator' : locator x <~> locator' x
      := equiv_adjointify
           locator_locator'
           locator'_locator
           locator'_locator_locator'
           locator_locator'_locator.

    Lemma nltqx_locates_left {q r : Q} (l' : locator' x) (ltqr : q < r)
      : ~ (' q < x) -> locates_left l' ltqr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
~ (' q < x) -> locates_left l' ltqr
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
~ (' q < x) -> locates_left l' ltqr
      assert (f := locates_right_true l' ltqr).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
f:locates_right l' ltqr -> ' q < x
~ (' q < x) -> locates_left l' ltqr
      exact (not_contrapositive f).
    Qed.

    Lemma ltxq_locates_left {q r : Q} (l' : locator' x) (ltqr : q < r)
      : x < ' q -> locates_left l' ltqr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
x < ' q -> locates_left l' ltqr
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
x < ' q -> locates_left l' ltqr
      intros ltxq.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
ltxq:x < ' q
locates_left l' ltqr apply nltqx_locates_left.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
ltxq:x < ' q
~ (' q < x)
      apply lt_flip; assumption.
    Qed.

    Lemma nltxr_locates_right {q r : Q} (l' : locator' x) (ltqr : q < r)
      : ~ (x < ' r) -> locates_right l' ltqr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
~ (x < ' r) -> locates_right l' ltqr
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
~ (x < ' r) -> locates_right l' ltqr
      intros nltxr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
nltxr:~ (x < ' r)
locates_right l' ltqr
      apply stable.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
nltxr:~ (x < ' r)
~~ locates_right l' ltqr
      assert (f := locates_right_false l' ltqr).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
nltxr:~ (x < ' r)
f:~ locates_right l' ltqr -> x < ' r
~~ locates_right l' ltqr
      exact (not_contrapositive f nltxr).
    Qed.

    Lemma ltrx_locates_right {q r : Q} (l' : locator' x) (ltqr : q < r)
      : 'r < x -> locates_right l' ltqr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
' r < x -> locates_right l' ltqr
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
' r < x -> locates_right l' ltqr
      intros ltrx.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
ltrx:' r < x
locates_right l' ltqr apply nltxr_locates_right.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
ltrx:' r < x
~ (x < ' r)
      apply lt_flip; assumption.
    Qed.

  End logic2.

  Definition ltQnegQ (q : Q) (eps : Qpos Q) : q - 'eps < q.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
q:Q
eps:Qpos Q
q - ' eps < q
  Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
q:Q
eps:Qpos Q
q - ' eps < q
    apply (pos_minus_lt_compat_r q ('eps)), eps.
  Qed.

  Local Open Scope mc_scope.

  Definition ltQposQ (q : Q) (eps : Qpos Q) : q < q + 'eps.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
q:Q
eps:Qpos Q
q < q + ' eps
  Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
q:Q
eps:Qpos Q
q < q + ' eps
    apply (pos_plus_lt_compat_r q ('eps)), eps.
  Qed.

  Section bounds.
    (* Given a real with a locator, we can find (integer) bounds. *)
    Context {x : F}
            (l : locator x).

    Definition ltN1 (q : Q) : q - 1 < q := ltQnegQ q 1.

    Definition P_lower (q : Q) : Type := locates_right l (ltN1 q).

    Definition P_lower_prop {k} : IsHProp (P_lower k).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
k:Q
IsHProp (P_lower k)
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
k:Q
IsHProp (P_lower k)
      apply _.
    Qed.

    Definition ltxN1 : x - 1 < x := (fst (pos_minus_lt_compat_r x 1) lt_0_1).

    Definition P_lower_inhab : hexists (fun q => P_lower q).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hexists (fun q : Q => P_lower q)
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hexists (fun q : Q => P_lower q)
      assert (hqlt : hexists (fun q => ' q < x)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hexists (fun q : Q => ' q < x)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun q : Q => ' q < x)
hexists (fun q : Q => P_lower q)
      {
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hexists (fun q : Q => ' q < x)
        assert (hex := archimedean_property Q F (x-1) x ltxN1).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hex:hexists (fun q : Q => x - 1 < ' q < x)
hexists (fun q : Q => ' q < x)
        refine (Trunc_rec _ hex); intros hex'.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hex:hexists (fun q : Q => x - 1 < ' q < x)
hex':{q : Q & x - 1 < ' q < x}
hexists (fun q : Q => ' q < x)
        apply tr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hex:hexists (fun q : Q => x - 1 < ' q < x)
hex':{q : Q & x - 1 < ' q < x}
{q : Q & ' q < x}
        destruct hex' as [q [ltx1q ltqx]]; exists q; assumption.
      }
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun q : Q => ' q < x)
hexists (fun q : Q => P_lower q)
      refine (Trunc_rec _ hqlt); intros hqlt'.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun q : Q => ' q < x)
hqlt':{q : Q & ' q < x}
hexists (fun q : Q => P_lower q)
      induction hqlt' as [q lt].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy canonical_names.lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun q : Q => ' q < x)
q:Q
lt:' q < x
hexists (fun q : Q => P_lower q)
      apply tr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy canonical_names.lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun q : Q => ' q < x)
q:Q
lt:' q < x
{q : Q & P_lower q}
      exists q.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy canonical_names.lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun q : Q => ' q < x)
q:Q
lt:' q < x
P_lower q
      unfold P_lower.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy canonical_names.lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun q : Q => ' q < x)
q:Q
lt:' q < x
locates_right l (ltN1 q)
      apply ltrx_locates_right; assumption.
    Qed.

    Definition lower_bound : {q : Q | ' q < x}.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
{q : Q & ' q < x}
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
{q : Q & ' q < x}
      assert (qP_lower : {q : Q | P_lower q})
        by refine (minimal_n_alt_type Q Q_eq P_lower _ P_lower_inhab).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
qP_lower:{q : Q & P_lower q}
{q : Q & ' q < x}
      destruct qP_lower as [q Pq].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q:Q
Pq:P_lower q
{q : Q & ' q < x}
      exists (q - 1).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q:Q
Pq:P_lower q
' (q - 1) < x
      unfold P_lower in Pq.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q:Q
Pq:locates_right l (ltN1 q)
' (q - 1) < x simpl in *.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q:Q
Pq:is_inl (l (q - 1) q (ltN1 q))
' (q - 1) < x
      apply (un_inl _ Pq).
    Qed.

    Definition lt1N (r : Q) : r < r + 1 := ltQposQ r 1.

    Definition P_upper (r : Q) : DHProp := locates_left l (lt1N r).

    Definition P_upper_prop {k} : IsHProp (P_upper k).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
k:Q
IsHProp (P_upper k)
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
k:Q
IsHProp (P_upper k)
      apply _.
    Qed.

    Definition ltx1N : x < x + 1 := (fst (pos_plus_lt_compat_r x 1) lt_0_1).

    Definition P_upper_inhab : hexists (fun r => P_upper r).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hexists (fun r : Q => P_upper r)
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hexists (fun r : Q => P_upper r)
      assert (hqlt : hexists (fun r => x < ' r)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hexists (fun r : Q => x < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun r : Q => x < ' r)
hexists (fun r : Q => P_upper r)
      {
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hexists (fun r : Q => x < ' r)
        assert (hex := archimedean_property Q F x (x+1) ltx1N).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hex:hexists (fun q : Q => x < ' q < x + 1)
hexists (fun r : Q => x < ' r)
        refine (Trunc_rec _ hex); intros hex'.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hex:hexists (fun q : Q => x < ' q < x + 1)
hex':{q : Q & x < ' q < x + 1}
hexists (fun r : Q => x < ' r)
        apply tr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hex:hexists (fun q : Q => x < ' q < x + 1)
hex':{q : Q & x < ' q < x + 1}
{r : Q & x < ' r}
        destruct hex' as [r [ltxr ltrx1]]; exists r; assumption.
      }
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun r : Q => x < ' r)
hexists (fun r : Q => P_upper r)
      refine (Trunc_rec _ hqlt); intros hqlt'.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun r : Q => x < ' r)
hqlt':{r : Q & x < ' r}
hexists (fun r : Q => P_upper r)
      induction hqlt' as [r lt].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy canonical_names.lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun r : Q => x < ' r)
r:Q
lt:x < ' r
hexists (fun r : Q => P_upper r)
      apply tr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy canonical_names.lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun r : Q => x < ' r)
r:Q
lt:x < ' r
{r : Q & P_upper r}
      exists r.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy canonical_names.lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun r : Q => x < ' r)
r:Q
lt:x < ' r
P_upper r
      unfold P_upper.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy canonical_names.lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun r : Q => x < ' r)
r:Q
lt:x < ' r
locates_left l (lt1N r)
      apply ltxq_locates_left; assumption.
    Qed.

    Definition upper_bound : {r : Q | x < ' r}.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
{r : Q & x < ' r}
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
{r : Q & x < ' r}
      assert (rP_upper : {r : Q | P_upper r})
        by refine (minimal_n_alt_type Q Q_eq P_upper _ P_upper_inhab).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
rP_upper:{r : Q & P_upper r}
{r : Q & x < ' r}
      destruct rP_upper as [r Pr].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
r:Q
Pr:P_upper r
{r : Q & x < ' r}
      exists (r + 1).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
r:Q
Pr:P_upper r
x < ' (r + 1)
      unfold P_upper in Pr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
r:Q
Pr:locates_left l (lt1N r)
x < ' (r + 1) simpl in *.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
r:Q
Pr:~ is_inl (l r (r + 1) (lt1N r))
x < ' (r + 1)
      destruct (l r (r + 1) (lt1N r)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
r:Q
l0:' r < x
Pr:~ is_inl (inl l0)
x < ' (r + 1)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
r:Q
l0:x < ' (r + 1)
Pr:~ is_inl (inr l0)
x < ' (r + 1)
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
r:Q
l0:' r < x
Pr:~ is_inl (inl l0)
x < ' (r + 1) simpl in Pr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
r:Q
l0:' r < x
Pr:~ Unit
x < ' (r + 1) destruct (Pr tt).
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
r:Q
l0:x < ' (r + 1)
Pr:~ is_inl (inr l0)
x < ' (r + 1) assumption.
    Qed.

    Instance inc_N_Q : Cast nat Q := naturals_to_semiring nat Q.

    Instance inc_fin_N {n} : Cast (Fin n) nat := fin_to_nat.

    Lemma tight_bound (epsilon : Qpos Q) : {u : Q | ' u < x < ' (u + ' epsilon)}.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
{u : Q & ' u < x < ' (u + ' epsilon)}
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
{u : Q & ' u < x < ' (u + ' epsilon)}
      destruct
        lower_bound as [q ltqx]
      , upper_bound as [r ltxr]
      , (round_up_strict Q ((3/'epsilon)*(r-q))) as [n lt3rqn].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
{u : Q & ' u < x < ' (u + ' epsilon)}
      assert (lt0 : 0 < 'epsilon / 3).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
0 < ' epsilon / 3
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3
{u : Q & ' u < x < ' (u + ' epsilon)}
      {
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
0 < ' epsilon / 3
        apply pos_mult.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
0 < ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
0 < / 3
        -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
0 < ' epsilon apply epsilon.
        -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
0 < / 3 apply pos_dec_recip_compat, lt_0_3.
      }
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3
{u : Q & ' u < x < ' (u + ' epsilon)}
      assert (lt0' : 0 < 3 / ' epsilon).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3
0 < 3 / ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3
lt0':0 < 3 / ' epsilon
{u : Q & ' u < x < ' (u + ' epsilon)}
      {
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3
0 < 3 / ' epsilon
        apply pos_mult.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3
0 < 3
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3
0 < / ' epsilon
        -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3
0 < 3 apply lt_0_3.
        -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3
0 < / ' epsilon apply pos_dec_recip_compat, epsilon.
      }
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3
lt0':0 < 3 / ' epsilon
{u : Q & ' u < x < ' (u + ' epsilon)}
      assert (ap30 : (3 : Q) <> 0)
        by apply lt_ne_flip, lt_0_3.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
{u : Q & ' u < x < ' (u + ' epsilon)}
      clear - l q ltqx r ltxr n lt3rqn lt0' ap30 Qtriv Qdec_paths H cast_pres_ordering.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
{u : Q & ' u < x < ' (u + ' epsilon)}
      assert (ltn3eps : r < q + ' n * ' epsilon / 3).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
r < q + ' n * ' epsilon / 3
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
{u : Q & ' u < x < ' (u + ' epsilon)}
      {
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
r < q + ' n * ' epsilon / 3
        rewrite (commutativity q (' n * ' epsilon / 3)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
r < ' n * ' epsilon / 3 + q
        apply flip_lt_minus_l.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
r - q < ' n * ' epsilon / 3
        apply (pos_mult_reflect_r (3 / ' epsilon) lt0').
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
(r - q) * (3 / ' epsilon) <
' n * ' epsilon / 3 * (3 / ' epsilon)
        rewrite (commutativity (r-q) (3 / ' epsilon)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
3 / ' epsilon * (r - q) <
' n * ' epsilon / 3 * (3 / ' epsilon)
        rewrite <- (associativity ('n) ('epsilon) (/3)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
3 / ' epsilon * (r - q) <
' n * (' epsilon / 3) * (3 / ' epsilon)
        rewrite <- (associativity ('n) (' epsilon / 3) (3 / ' epsilon)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
3 / ' epsilon * (r - q) <
' n * (' epsilon / 3 * (3 / ' epsilon))
        rewrite <- (associativity ('epsilon) (/3) (3/'epsilon)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
3 / ' epsilon * (r - q) <
' n * (' epsilon * (/ 3 * (3 / ' epsilon)))
        rewrite (associativity (/3) 3 (/'epsilon)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
3 / ' epsilon * (r - q) <
' n * (' epsilon * (/ 3 * 3 / ' epsilon))
        rewrite (commutativity (/3) 3).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
3 / ' epsilon * (r - q) <
' n * (' epsilon * (3 / 3 / ' epsilon))
        rewrite (dec_recip_inverse 3 ap30).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
3 / ' epsilon * (r - q) <
' n * (' epsilon * (1 / ' epsilon))
        rewrite mult_1_l.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
3 / ' epsilon * (r - q) <
' n * (' epsilon / ' epsilon)
        assert (apepsilon0 : 'epsilon <> 0)
          by apply lt_ne_flip, epsilon.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
apepsilon0:' epsilon <> 0
3 / ' epsilon * (r - q) <
' n * (' epsilon / ' epsilon)
        rewrite (dec_recip_inverse ('epsilon) apepsilon0).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
apepsilon0:' epsilon <> 0
3 / ' epsilon * (r - q) < ' n * 1
        rewrite mult_1_r.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
apepsilon0:' epsilon <> 0
3 / ' epsilon * (r - q) < ' n
        assumption.
      }
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
{u : Q & ' u < x < ' (u + ' epsilon)}
      set (grid (k : Fin n.+3) := q + (' (' k) - 1)*('epsilon/3) : Q).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
{u : Q & ' u < x < ' (u + ' epsilon)}
      assert (lt_grid : forall k : Fin _, grid (fin_incl k) < grid (fsucc k)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
{u : Q & ' u < x < ' (u + ' epsilon)}
      {
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
        intros k.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
grid (fin_incl k) < grid (fsucc k) unfold grid.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
        change (' fin_incl k) with (fin_to_nat (fin_incl k));
          rewrite path_nat_fin_incl.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
q + (' fin_to_nat k - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
        change (' fsucc k) with (fin_to_nat (fsucc k));
          rewrite path_nat_fsucc.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
q + (' fin_to_nat k - 1) * (' epsilon / 3) <
q + (' ((fin_to_nat k).+1)%nat - 1) * (' epsilon / 3)
        assert (' (S (' k)) = (' (' k) + 1)) as ->.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
' ((' k).+1)%nat = ' (' k) + 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
q + (' fin_to_nat k - 1) * (' epsilon / 3) <
q + (' (' k) + 1 - 1) * (' epsilon / 3)
        {
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
' ((' k).+1)%nat = ' (' k) + 1
          rewrite S_nat_plus_1.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
' (' k + 1) = ' (' k) + 1
          rewrite (preserves_plus (' k) 1).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
' (' k) + ' 1 = ' (' k) + 1
          rewrite preserves_1.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
' (' k) + 1 = ' (' k) + 1
          reflexivity.
        }
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
q + (' fin_to_nat k - 1) * (' epsilon / 3) <
q + (' (' k) + 1 - 1) * (' epsilon / 3)
        assert (' (' k) + 1 - 1 = ' (' k) - 1 + 1) as ->.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
' (' k) + 1 - 1 = ' (' k) - 1 + 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
q + (' fin_to_nat k - 1) * (' epsilon / 3) <
q + (' (' k) - 1 + 1) * (' epsilon / 3)
        {
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
' (' k) + 1 - 1 = ' (' k) - 1 + 1
          rewrite <- (associativity _ 1 (-1)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
' (' k) + (1 - 1) = ' (' k) - 1 + 1
          rewrite (commutativity 1 (-1)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
' (' k) + (-1 + 1) = ' (' k) - 1 + 1
          rewrite (associativity _ (-1) 1).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
' (' k) - 1 + 1 = ' (' k) - 1 + 1
          reflexivity.
        }
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
q + (' fin_to_nat k - 1) * (' epsilon / 3) <
q + (' (' k) - 1 + 1) * (' epsilon / 3)
        assert (lt1 : ' (' k) - 1 < ' (' k) - 1 + 1)
          by apply pos_plus_lt_compat_r, lt_0_1.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
lt1:' (' k) - 1 < ' (' k) - 1 + 1
q + (' fin_to_nat k - 1) * (' epsilon / 3) <
q + (' (' k) - 1 + 1) * (' epsilon / 3)
        assert (lt2 : (' (' k) - 1) * (' epsilon / 3) < (' (' k) - 1 + 1) * (' epsilon / 3)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
lt1:' (' k) - 1 < ' (' k) - 1 + 1
(' (' k) - 1) * (' epsilon / 3) <
(' (' k) - 1 + 1) * (' epsilon / 3)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
lt1:' (' k) - 1 < ' (' k) - 1 + 1
lt2:(' (' k) - 1) * (' epsilon / 3) <
(' (' k) - 1 + 1) * (' epsilon / 3)
q + (' fin_to_nat k - 1) * (' epsilon / 3) <
q + (' (' k) - 1 + 1) * (' epsilon / 3)
        {
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
lt1:' (' k) - 1 < ' (' k) - 1 + 1
(' (' k) - 1) * (' epsilon / 3) <
(' (' k) - 1 + 1) * (' epsilon / 3)
          nrefine (pos_mult_lt_r ('epsilon/3) _ (' (' k) - 1) (' (' k) - 1 + 1) _); try apply _.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
lt1:' (' k) - 1 < ' (' k) - 1 + 1
' (' k) - 1 < ' (' k) - 1 + 1
          apply lt1.
        }
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
lt1:' (' k) - 1 < ' (' k) - 1 + 1
lt2:(' (' k) - 1) * (' epsilon / 3) <
(' (' k) - 1 + 1) * (' epsilon / 3)
q + (' fin_to_nat k - 1) * (' epsilon / 3) <
q + (' (' k) - 1 + 1) * (' epsilon / 3)
        apply pseudo_srorder_plus.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
lt1:' (' k) - 1 < ' (' k) - 1 + 1
lt2:(' (' k) - 1) * (' epsilon / 3) <
(' (' k) - 1 + 1) * (' epsilon / 3)
(' fin_to_nat k - 1) * (' epsilon / 3) <
(' (' k) - 1 + 1) * (' epsilon / 3)
        exact lt2.
      }
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
{u : Q & ' u < x < ' (u + ' epsilon)}
      set (P k := locates_right l (lt_grid k)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
{u : Q & ' u < x < ' (u + ' epsilon)}
      assert (left_true : P fin_zero).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
P fin_zero
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
{u : Q & ' u < x < ' (u + ' epsilon)}
      {
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
P fin_zero
        apply ltrx_locates_right.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
' grid (fsucc fin_zero) < x
        unfold grid.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
' (q + (' (' fsucc fin_zero) - 1) * (' epsilon / 3)) <
x
        change (' fsucc fin_zero) with (fin_to_nat (@fsucc (S n) fin_zero)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
' (q +
   (' fin_to_nat (fsucc fin_zero) - 1) *
   (' epsilon / 3)) < x
        rewrite path_nat_fsucc, path_nat_fin_zero.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
' (q + (' 1%nat - 1) * (' epsilon / 3)) < x
        rewrite (@preserves_1 nat Q _ _ _ _ _ _ _ _ _ _).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
' (q + (1 - 1) * (' epsilon / 3)) < x
        rewrite plus_negate_r.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
' (q + 0 * (' epsilon / 3)) < x
        rewrite mult_0_l.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
' (q + 0) < x
        rewrite plus_0_r.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
' q < x
        assumption.
      }
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
{u : Q & ' u < x < ' (u + ' epsilon)}
      assert (right_false : ~ P fin_last).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
~ P fin_last
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
right_false:~ P fin_last
{u : Q & ' u < x < ' (u + ' epsilon)}
      {
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
~ P fin_last
        apply ltxq_locates_left.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
x < ' grid (fin_incl fin_last)
        unfold grid.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
x <
' (q + (' (' fin_incl fin_last) - 1) * (' epsilon / 3))
        change (' fin_incl fin_last) with (fin_to_nat (@fin_incl (S (S n)) fin_last)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
x <
' (q +
   (' fin_to_nat (fin_incl fin_last) - 1) *
   (' epsilon / 3))
        rewrite path_nat_fin_incl, path_nat_fin_last.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
x < ' (q + (' (n.+1)%nat - 1) * (' epsilon / 3))
        rewrite S_nat_plus_1.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
x < ' (q + (' (n + 1) - 1) * (' epsilon / 3))
        rewrite (preserves_plus n 1).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
x < ' (q + (' n + ' 1 - 1) * (' epsilon / 3))
        rewrite (@preserves_1 nat Q _ _ _ _ _ _ _ _ _ _).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
x < ' (q + (' n + 1 - 1) * (' epsilon / 3))
        rewrite <- (associativity (' n) 1 (-1)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
x < ' (q + (' n + (1 - 1)) * (' epsilon / 3))
        rewrite plus_negate_r.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
x < ' (q + (' n + 0) * (' epsilon / 3))
        rewrite plus_0_r.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
x < ' (q + ' n * (' epsilon / 3))
        rewrite (associativity ('n) ('epsilon) (/3)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
x < ' (q + ' n * ' epsilon / 3)
        transitivity ('r).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
x < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
' r < ' (q + ' n * ' epsilon / 3)
        -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
x < ' r exact ltxr.
        -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
' r < ' (q + ' n * ' epsilon / 3) apply strictly_order_preserving; try trivial.
      }
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
right_false:~ P fin_last
{u : Q & ' u < x < ' (u + ' epsilon)}
      destruct (sperners_lemma_1d P left_true right_false) as [u [Pltux Pltxueps]].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
right_false:~ P fin_last
u:Fin n.+1
Pltux:P (fin_incl u)
Pltxueps:~ P (fsucc u)
{u : Q & ' u < x < ' (u + ' epsilon)}
      exists (grid (fin_incl (fin_incl u))).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
right_false:~ P fin_last
u:Fin n.+1
Pltux:P (fin_incl u)
Pltxueps:~ P (fsucc u)
' grid (fin_incl (fin_incl u)) < x <
' (grid (fin_incl (fin_incl u)) + ' epsilon)
      unfold P in Pltux, Pltxueps.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
right_false:~ P fin_last
u:Fin n.+1
Pltux:locates_right l (lt_grid (fin_incl u))
Pltxueps:~ locates_right l (lt_grid (fsucc u))
' grid (fin_incl (fin_incl u)) < x <
' (grid (fin_incl (fin_incl u)) + ' epsilon)
      split.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
right_false:~ P fin_last
u:Fin n.+1
Pltux:locates_right l (lt_grid (fin_incl u))
Pltxueps:~ locates_right l (lt_grid (fsucc u))
' grid (fin_incl (fin_incl u)) < x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
right_false:~ P fin_last
u:Fin n.+1
Pltux:locates_right l (lt_grid (fin_incl u))
Pltxueps:~ locates_right l (lt_grid (fsucc u))
x < ' (grid (fin_incl (fin_incl u)) + ' epsilon)
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
right_false:~ P fin_last
u:Fin n.+1
Pltux:locates_right l (lt_grid (fin_incl u))
Pltxueps:~ locates_right l (lt_grid (fsucc u))
' grid (fin_incl (fin_incl u)) < x apply (locates_right_true l (lt_grid (fin_incl u)) Pltux).
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) <
naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
right_false:~ P fin_last
u:Fin n.+1
Pltux:locates_right l (lt_grid (fin_incl u))
Pltxueps:~ locates_right l (lt_grid (fsucc u))
x < ' (grid (fin_incl (fin_incl u)) + ' epsilon) clear - Pltxueps Qtriv Qdec_paths ap30 cast_pres_ordering.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:(3 : Q) <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
x < ' (grid (fin_incl (fin_incl u)) + ' epsilon)
        set (ltxbla := locates_right_false l (lt_grid (fsucc u)) Pltxueps).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:(3 : Q) <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
grid (fin_incl k) < grid (fsucc k)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:=locates_right_false l (lt_grid (fsucc u))
  Pltxueps:x < ' grid (fsucc (fsucc u))
x < ' (grid (fin_incl (fin_incl u)) + ' epsilon)
        unfold grid in *.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:=locates_right_false l (lt_grid (fsucc u))
  Pltxueps:x <
' (q +
   (' (' fsucc (fsucc u)) - 1) *
   (' epsilon / 3))
x <
' (q +
   (' (' fin_incl (fin_incl u)) - 1) * (' epsilon / 3) +
   ' epsilon)
        change (' fin_incl (fin_incl u)) with (fin_to_nat (fin_incl (fin_incl u))).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:=locates_right_false l (lt_grid (fsucc u))
  Pltxueps:x <
' (q +
   (' (' fsucc (fsucc u)) - 1) *
   (' epsilon / 3))
x <
' (q +
   (' fin_to_nat (fin_incl (fin_incl u)) - 1) *
   (' epsilon / 3) + ' epsilon)
        rewrite path_nat_fin_incl, path_nat_fin_incl.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:=locates_right_false l (lt_grid (fsucc u))
  Pltxueps:x <
' (q +
   (' (' fsucc (fsucc u)) - 1) *
   (' epsilon / 3))
x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   ' epsilon)
        change (' fsucc (fsucc u)) with (fin_to_nat (fsucc (fsucc u))) in ltxbla.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:=locates_right_false l (lt_grid (fsucc u))
  Pltxueps:x <
' (q +
   (' fin_to_nat (fsucc (fsucc u)) - 1) *
   (' epsilon / 3))
x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   ' epsilon)
        rewrite path_nat_fsucc, path_nat_fsucc in ltxbla.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q +
   (' ((fin_to_nat u).+2)%nat - 1) *
   (' epsilon / 3))
x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   ' epsilon)
        rewrite S_nat_plus_1, S_nat_plus_1 in ltxbla.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q +
   (' (fin_to_nat u + 1 + 1) - 1) *
   (' epsilon / 3))
x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   ' epsilon)
        rewrite (preserves_plus (fin_to_nat u + 1) 1) in ltxbla.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q +
   (' (fin_to_nat u + 1) + ' 1 - 1) *
   (' epsilon / 3))
x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   ' epsilon)
        rewrite (preserves_plus (fin_to_nat u) 1) in ltxbla.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q +
   (' fin_to_nat u + ' 1 + ' 1 - 1) *
   (' epsilon / 3))
x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   ' epsilon)
        rewrite preserves_1 in ltxbla.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q +
   (' fin_to_nat u + 1 + 1 - 1) *
   (' epsilon / 3))
x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   ' epsilon)
        rewrite <- (associativity (' fin_to_nat u) 1 1) in ltxbla.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q +
   (' fin_to_nat u + 2 - 1) * (' epsilon / 3))
x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   ' epsilon)
        rewrite <- (associativity (' fin_to_nat u) 2 (-1)) in ltxbla.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q +
   (' fin_to_nat u + (2 - 1)) *
   (' epsilon / 3))
x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   ' epsilon)
        rewrite (commutativity 2 (-1)) in ltxbla.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q +
   (' fin_to_nat u + (-1 + 2)) *
   (' epsilon / 3))
x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   ' epsilon)
        rewrite (associativity (' fin_to_nat u) (-1) 2) in ltxbla.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q +
   (' fin_to_nat u - 1 + 2) * (' epsilon / 3))
x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   ' epsilon)
        rewrite plus_mult_distr_r in ltxbla.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q +
   ((' fin_to_nat u - 1) * (' epsilon / 3) +
    2 * (' epsilon / 3)))
x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   ' epsilon)
        rewrite (associativity q ((' fin_to_nat u - 1) * (' epsilon / 3)) (2 * (' epsilon / 3))) in ltxbla.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   2 * (' epsilon / 3))
x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   ' epsilon)
        refine (transitivity ltxbla _).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   2 * (' epsilon / 3))
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   2 * (' epsilon / 3)) <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   ' epsilon)
        apply strictly_order_preserving; try apply _.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   2 * (' epsilon / 3))
q + (' fin_to_nat u - 1) * (' epsilon / 3) +
2 * (' epsilon / 3) <
q + (' fin_to_nat u - 1) * (' epsilon / 3) + ' epsilon
        apply pseudo_srorder_plus.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   2 * (' epsilon / 3))
2 * (' epsilon / 3) < ' epsilon
        rewrite (associativity 2 ('epsilon) (/3)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   2 * (' epsilon / 3))
2 * ' epsilon / 3 < ' epsilon
        rewrite (commutativity 2 ('epsilon)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   2 * (' epsilon / 3))
' epsilon * 2 / 3 < ' epsilon
        rewrite <- (mult_1_r ('epsilon)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   2 * (' epsilon / 3))
' epsilon * 1 * 2 / 3 < ' epsilon * 1
        rewrite <- (associativity ('epsilon) 1 2).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   2 * (' epsilon / 3))
' epsilon * (1 * 2) / 3 < ' epsilon * 1
        rewrite (mult_1_l 2).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   2 * (' epsilon / 3))
' epsilon * 2 / 3 < ' epsilon * 1
        rewrite <- (associativity ('epsilon) 2 (/3)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   2 * (' epsilon / 3))
' epsilon * (2 / 3) < ' epsilon * 1
        apply pos_mult_lt_l.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   2 * (' epsilon / 3))
PropHolds (0 < ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   2 * (' epsilon / 3))
2 / 3 < 1
        +
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   2 * (' epsilon / 3))
PropHolds (0 < ' epsilon) apply epsilon.
        +
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   2 * (' epsilon / 3))
2 / 3 < 1 nrefine (pos_mult_reflect_r (3 : Q) lt_0_3 _ _ _); try apply _.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   2 * (' epsilon / 3))
2 / 3 * 3 < 1 * 3
          rewrite <- (associativity 2 (/3) 3).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   2 * (' epsilon / 3))
2 * (/ 3 * 3) < 1 * 3
          rewrite (commutativity (/3) 3).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   2 * (' epsilon / 3))
2 * (3 / 3) < 1 * 3
          rewrite (dec_recip_inverse (3 : Q) ap30).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   2 * (' epsilon / 3))
2 * 1 < 1 * 3
          rewrite (mult_1_r 2).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   2 * (' epsilon / 3))
2 < 1 * 3
          rewrite (mult_1_l 3).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 =>
q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2,
q + (' (' fin_incl k) - 1) * (' epsilon / 3) <
q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x <
' (q + (' fin_to_nat u - 1) * (' epsilon / 3) +
   2 * (' epsilon / 3))
2 < 3
          exact lt_2_3.
    Qed.

  End bounds.

  Section arch_struct.

    Context {x y : F}
            (l : locator x)
            (m : locator y)
            (ltxy : x < y).

    Definition P (qeps' : Q * Qpos Q) : Type :=
      match qeps' with
      | (q' , eps') =>
        (prod
           (locates_left l (ltQnegQ q' eps'))
           (locates_right m (ltQposQ q' eps')))
      end.

    Definition P_isHProp qeps' : IsHProp (P qeps').
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
qeps':(Q * Qpos Q)%type
IsHProp (P qeps')
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
qeps':(Q * Qpos Q)%type
IsHProp (P qeps')
      destruct qeps' as [q eps'].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
q:Q
eps':Qpos Q
IsHProp (P (q, eps'))
      apply istrunc_prod.
    Qed.

    Definition P_dec qeps' : Decidable (P qeps').
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
qeps':(Q * Qpos Q)%type
Decidable (P qeps')
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
qeps':(Q * Qpos Q)%type
Decidable (P qeps')
      destruct qeps' as [q eps'].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
q:Q
eps':Qpos Q
Decidable (P (q, eps'))
      unfold P.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
q:Q
eps':Qpos Q
Decidable
  (locates_left l (ltQnegQ q eps') *
   locates_right m (ltQposQ q eps'))
      apply _.
    Qed.

    Definition P_inhab : hexists P.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hexists P
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hexists P
      assert (hs := (archimedean_property Q F x y ltxy)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
hexists P
      refine (Trunc_ind _ _ hs); intros [s [ltxs ltsy]].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
hexists P
      assert (ht := (archimedean_property Q F ('s) y ltsy)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
hexists P
      refine (Trunc_ind _ _ ht); intros [t [ltst' ltty]].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
hexists P
      set (q := (t + s) / 2).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
hexists P
      assert (ltst : s < t).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
s < t
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
hexists P
      {
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
s < t
        Existing Instance full_pseudo_order_reflecting.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
s < t
        refine (strictly_order_reflecting _ _ _ ltst').
      }
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
hexists P
      set (epsilon := (Qpos_diff s t ltst) / 2).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
hexists P
      apply tr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
{x : _ & P x}
      exists (q, epsilon).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
P (q, epsilon)
      unfold P; split.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
locates_left l (ltQnegQ q epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
locates_right m (ltQposQ q epsilon)
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
locates_left l (ltQnegQ q epsilon) apply ltxq_locates_left.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
x < ' (q - ' epsilon)
        assert (q - ' epsilon = s) as ->.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
q - ' epsilon = s
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
x < ' s
        {
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
q - ' epsilon = s
          unfold q; cbn.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
(t + s) / 2 - (t - s) / 2 = s
          rewrite <- path_avg_split_diff_l.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
s + (t - s) / 2 - (t - s) / 2 = s
          rewrite <- (plus_assoc s ((t-s)/2) (-((t-s)/2))).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
s + ((t - s) / 2 - (t - s) / 2) = s
          rewrite plus_negate_r.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
s + 0 = s
          rewrite plus_0_r.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
s = s
          reflexivity.
        }
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
x < ' s
        assumption.
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
locates_right m (ltQposQ q epsilon) apply ltrx_locates_right.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
' (q + ' epsilon) < y
        assert (q + ' epsilon = t) as ->.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
q + ' epsilon = t
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
' t < y
        {
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
q + ' epsilon = t
          unfold q; cbn.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
(t + s) / 2 + (t - s) / 2 = t
          rewrite <- path_avg_split_diff_r.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
t - (t - s) / 2 + (t - s) / 2 = t
          rewrite <- (plus_assoc t (-((t-s)/2)) ((t-s)/2)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
t + (- ((t - s) / 2) + (t - s) / 2) = t
          rewrite plus_negate_l.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
t + 0 = t
          rewrite plus_0_r.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
t = t
          reflexivity.
        }
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
' t < y
        assumption.
    Qed.

    Definition archimedean_structure : {q : Q | x < 'q < y}.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
{q : Q & x < ' q < y}
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
{q : Q & x < ' q < y}
      assert (R : sig P).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
{x : _ & P x}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
R:{x : _ & P x}
{q : Q & x < ' q < y}
      {
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
{x : _ & P x}
        apply minimal_n_alt_type.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
nat <~> Q * Qpos Q
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
forall x : Q * Qpos Q, IsHProp (P x)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
forall x : Q * Qpos Q, Decidable (P x)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hexists (fun x : Q * Qpos Q => P x)
        -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
nat <~> Q * Qpos Q apply QQpos_eq.
        -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
forall x : Q * Qpos Q, IsHProp (P x) apply P_isHProp.
        -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
forall x : Q * Qpos Q, Decidable (P x) apply P_dec.
        -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hexists (fun x : Q * Qpos Q => P x) apply P_inhab.
      }
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
R:{x : _ & P x}
{q : Q & x < ' q < y}
      unfold P in R.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
R:{qeps' : Q * Qpos Q &
(locates_left l (ltQnegQ (fst qeps') (snd qeps')) *
 locates_right m (ltQposQ (fst qeps') (snd qeps')))%type}
{q : Q & x < ' q < y}
      destruct R as [[q eps] [lleft mright]].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
q:Q
eps:Qpos Q
lleft:locates_left l (ltQnegQ q eps)
mright:locates_right m (ltQposQ q eps)
{q : Q & x < ' q < y}
      exists q; split.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
q:Q
eps:Qpos Q
lleft:locates_left l (ltQnegQ q eps)
mright:locates_right m (ltQposQ q eps)
x < ' q
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
q:Q
eps:Qpos Q
lleft:locates_left l (ltQnegQ q eps)
mright:locates_right m (ltQposQ q eps)
' q < y
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
q:Q
eps:Qpos Q
lleft:locates_left l (ltQnegQ q eps)
mright:locates_right m (ltQposQ q eps)
x < ' q nrefine (locates_right_false l _ lleft).
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
q:Q
eps:Qpos Q
lleft:locates_left l (ltQnegQ q eps)
mright:locates_right m (ltQposQ q eps)
' q < y nrefine (locates_right_true  m _ mright).
    Qed.

  End arch_struct.

  Section unary_ops.

    Context {x : F}
            (l : locator x).

    Definition locator_minus : locator (-x).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
locator (- x)
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
locator (- x)
      intros q r ltqr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
((' q < - x) + (- x < ' r))%type
      assert (ltnrnq := snd (flip_lt_negate q r) ltqr : -r < -q).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q
((' q < - x) + (- x < ' r))%type
      destruct (l _ _ ltnrnq) as [ltnrx|ltxnq].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q
ltnrx:' (- r) < x
((' q < - x) + (- x < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q
ltxnq:x < ' (- q)
((' q < - x) + (- x < ' r))%type
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q
ltnrx:' (- r) < x
((' q < - x) + (- x < ' r))%type apply inr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q
ltnrx:' (- r) < x
- x < ' r apply char_minus_left.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q
ltnrx:' (- r) < x
- ' r < x rewrite <- preserves_negate.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q
ltnrx:' (- r) < x
' (- r) < x assumption.
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q
ltxnq:x < ' (- q)
((' q < - x) + (- x < ' r))%type apply inl.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q
ltxnq:x < ' (- q)
' q < - x apply char_minus_right.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q
ltxnq:x < ' (- q)
x < - ' q rewrite <- preserves_negate.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q
ltxnq:x < ' (- q)
x < ' (- q) assumption.
    Qed.

    Section recip_pos.
      Context (xpos : 0 < x).

      Definition recip_nu := positive_apart_zero x xpos.

      Definition locator_recip_pos : locator (// (x ; recip_nu)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
locator (// (x; recip_nu))
      Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
locator (// (x; recip_nu))
        assert (recippos : 0 < // (x ; recip_nu))
          by apply pos_recip_compat.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
locator (// (x; recip_nu))
        intros q r ltqr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type destruct (trichotomy _ q 0) as [qneg|[qzero|qpos]].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qneg:q < 0
((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qzero:q = 0
((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
        +
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qneg:q < 0
((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type apply inl.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qneg:q < 0
' q < // (x; recip_nu) refine (transitivity _ _).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qneg:q < 0
' q < ?Goal1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qneg:q < 0
?Goal1 < // (x; recip_nu)
          *
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qneg:q < 0
' q < ?Goal1 apply (strictly_order_preserving _).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qneg:q < 0
q < ?y exact qneg.
          *
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qneg:q < 0
' 0 < // (x; recip_nu) rewrite preserves_0; assumption.
        +
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qzero:q = 0
((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type apply inl.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qzero:q = 0
' q < // (x; recip_nu) rewrite qzero, preserves_0; assumption.
        +
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type assert (qap0 : q ≶ 0)
            by apply (pseudo_order_lt_apart_flip _ _ qpos).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
          assert (rap0 : r ≶ 0).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
r ≶ 0
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
          {
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
r ≶ 0
            refine (pseudo_order_lt_apart_flip _ _ _).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
0 < r
            apply (transitivity qpos ltqr).
          }
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
          assert (ltrrrq : / r < / q)
            by (apply flip_lt_dec_recip; assumption).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
          destruct (l (/r) (/q) ltrrrq) as [ltrrx|ltxrq].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltrrx:' (/ r) < x
((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltxrq:x < ' (/ q)
((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
          *
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltrrx:' (/ r) < x
((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type apply inr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltrrx:' (/ r) < x
// (x; recip_nu) < ' r
            assert (rpos : 0 < r)
              by (transitivity q; assumption).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltrrx:' (/ r) < x
rpos:0 < r
// (x; recip_nu) < ' r
            assert (rpos' : 0 < ' r).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltrrx:' (/ r) < x
rpos:0 < r
0 < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltrrx:' (/ r) < x
rpos:0 < r
rpos':0 < ' r
// (x; recip_nu) < ' r
            {
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltrrx:' (/ r) < x
rpos:0 < r
0 < ' r
              rewrite <- (@preserves_0 Q F _ _ _ _ _ _ _ _ _ _).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltrrx:' (/ r) < x
rpos:0 < r
qinc 0 < ' r
              apply strictly_order_preserving; try apply _; assumption.
            }
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltrrx:' (/ r) < x
rpos:0 < r
rpos':0 < ' r
// (x; recip_nu) < ' r
            rewrite (dec_recip_to_recip r (positive_apart_zero ('r) rpos')) in ltrrx.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
rpos':0 < ' r
ltrrx:// (' r; positive_apart_zero (' r) rpos') < x
rpos:0 < r
// (x; recip_nu) < ' r
            assert (ltxrr := flip_lt_recip_l x ('r) rpos' ltrrx).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
rpos':0 < ' r
ltrrx:// (' r; positive_apart_zero (' r) rpos') < x
rpos:0 < r
ltxrr:let apx0 :=
  positive_apart_zero x
    (transitivity
       (pos_recip_compat (' r) rpos') ltrrx)
  in
// (x; apx0) < ' r
// (x; recip_nu) < ' r
            cbn in ltxrr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
rpos':0 < ' r
ltrrx:// (' r; positive_apart_zero (' r) rpos') < x
rpos:0 < r
ltxrr://
(x;
positive_apart_zero x
  (strictorder_trans 0
     (//
      (' r; positive_apart_zero (' r) rpos'))
     x (pos_recip_compat (' r) rpos') ltrrx)) <
' r
// (x; recip_nu) < ' r
            rewrite
              (recip_irrelevant
                 x
                 (positive_apart_zero
                    x
                    (transitivity (pos_recip_compat (' r) rpos') ltrrx)) recip_nu) in ltxrr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
rpos':0 < ' r
ltrrx:// (' r; positive_apart_zero (' r) rpos') < x
rpos:0 < r
ltxrr:// (x; recip_nu) < ' r
// (x; recip_nu) < ' r
            exact ltxrr.
          *
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltxrq:x < ' (/ q)
((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type apply inl.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltxrq:x < ' (/ q)
' q < // (x; recip_nu)
            assert (qpos' : 0 < ' q).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltxrq:x < ' (/ q)
0 < ' q
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltxrq:x < ' (/ q)
qpos':0 < ' q
' q < // (x; recip_nu)
            {
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltxrq:x < ' (/ q)
0 < ' q
              rewrite <- (@preserves_0 Q F _ _ _ _ _ _ _ _ _ _).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltxrq:x < ' (/ q)
qinc 0 < ' q
              apply strictly_order_preserving; try apply _; assumption.
            }
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltxrq:x < ' (/ q)
qpos':0 < ' q
' q < // (x; recip_nu)
            rewrite (dec_recip_to_recip q (positive_apart_zero ('q) qpos')) in ltxrq.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
qpos':0 < ' q
ltxrq:x < // (' q; positive_apart_zero (' q) qpos')
' q < // (x; recip_nu)
            assert (ltrqx := flip_lt_recip_r ('q) x qpos' xpos ltxrq).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
qpos':0 < ' q
ltxrq:x < // (' q; positive_apart_zero (' q) qpos')
ltrqx:' q < // (x; positive_apart_zero x xpos)
' q < // (x; recip_nu)
            rewrite (recip_irrelevant x (positive_apart_zero x xpos) recip_nu) in ltrqx.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
qpos':0 < ' q
ltxrq:x < // (' q; positive_apart_zero (' q) qpos')
ltrqx:' q < // (x; recip_nu)
' q < // (x; recip_nu)
            exact ltrqx.
      Qed.
    End recip_pos.

  End unary_ops.

  Section recip_neg.
    Context {x : F}
            (l : locator x)
            (xneg : x < 0).

    Definition recip_neg_nu := negative_apart_zero x xneg.

    Definition locator_recip_neg : locator (// (x ; recip_neg_nu)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xneg:x < 0
locator (// (x; recip_neg_nu))
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xneg:x < 0
locator (// (x; recip_neg_nu))
      assert (negxpos : 0 < (-x))
        by (apply flip_neg_negate; assumption).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xneg:x < 0
negxpos:0 < - x
locator (// (x; recip_neg_nu))
      assert (l' := locator_minus (locator_recip_pos (locator_minus l) negxpos)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xneg:x < 0
negxpos:0 < - x
l':locator (- (// (- x; recip_nu negxpos)))
locator (// (x; recip_neg_nu))
      rewrite (recip_negate (-x)) in l'.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xneg:x < 0
negxpos:0 < - x
l':locator (// negate_apart (- x; recip_nu negxpos))
locator (// (x; recip_neg_nu))
      unfold negate_apart in l'.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xneg:x < 0
negxpos:0 < - x
l':locator
  (//
   (- - x; apart_negate (- x) (recip_nu negxpos)))
locator (// (x; recip_neg_nu))
      rewrite
        (recip_proper_alt
           (- - x)
           x
           (apart_negate (- x) (positive_apart_zero (- x) negxpos)) recip_neg_nu)
        in l'.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xneg:x < 0
negxpos:0 < - x
l':locator (// (x; recip_neg_nu))
locator (// (x; recip_neg_nu))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xneg:x < 0
negxpos:0 < - x
l':locator
  (//
   (- - x; apart_negate (- x) (recip_nu negxpos)))
- - x = x
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xneg:x < 0
negxpos:0 < - x
l':locator (// (x; recip_neg_nu))
locator (// (x; recip_neg_nu)) assumption.
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xneg:x < 0
negxpos:0 < - x
l':locator
  (//
   (- - x; apart_negate (- x) (recip_nu negxpos)))
- - x = x apply negate_involutive.
    Qed.

  End recip_neg.

  Section unary_ops2.

    Context {x : F}
            (l : locator x)
            (nu : x ≶ 0).

    Definition locator_recip : locator (// (x ; nu)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
nu:x ≶ 0
locator (// (x; nu))
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
nu:x ≶ 0
locator (// (x; nu))
      destruct (fst (apart_iff_total_lt x 0) nu) as [xneg|xpos].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
nu:x ≶ 0
xneg:x < 0
locator (// (x; nu))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
nu:x ≶ 0
xpos:0 < x
locator (// (x; nu))
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
nu:x ≶ 0
xneg:x < 0
locator (// (x; nu)) set (l' := locator_recip_neg l xneg).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
nu:x ≶ 0
xneg:x < 0
l':=locator_recip_neg l xneg:locator (// (x; recip_neg_nu xneg))
locator (// (x; nu))
        rewrite (recip_proper_alt x x (negative_apart_zero x xneg) nu) in l';
          try reflexivity; exact l'.
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
nu:x ≶ 0
xpos:0 < x
locator (// (x; nu)) set (l' := locator_recip_pos l xpos).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
nu:x ≶ 0
xpos:0 < x
l':=locator_recip_pos l xpos:locator (// (x; recip_nu xpos))
locator (// (x; nu))
        rewrite (recip_proper_alt x x (positive_apart_zero x xpos) nu) in l';
          try reflexivity; exact l'.
    Qed.

  End unary_ops2.

  Section binary_ops.

    Context {x y : F}
            (l : locator x)
            (m : locator y).

    (** TODO the following two should be proven in Classes/orders/archimedean.v *)
    Context (char_plus_left : forall (q : Q) (x y : F),
                ' q < x + y <-> hexists (fun s : Q => (' s < x) /\ (' (q - s) < y)))
            (char_plus_right : forall (r : Q) (x y : F),
                x + y < ' r <-> hexists (fun t : Q => (x < ' t) /\ (y < ' (r - t)))).

    Definition locator_plus : locator (x + y).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
locator (x + y)
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
locator (x + y)
      intros q r ltqr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
      set (epsilon := (Qpos_diff q r ltqr) / 2).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
      assert (q+'epsilon=r-'epsilon)
        by (rewrite path_avg_split_diff_l, path_avg_split_diff_r; reflexivity).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
      destruct (tight_bound m epsilon) as [u [ltuy ltyuepsilon]].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
      set (s := q-u).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
      assert (qsltx : 'q-'s<y).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
' q - ' s < y
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
      {
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
' q - ' s < y
        unfold s.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
' q - ' (q - u) < y
        rewrite (preserves_plus q (-u)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
' q - (' q + ' (- u)) < y
        rewrite negate_plus_distr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
' q + (- ' q - ' (- u)) < y
        rewrite (associativity ('q) (-'q) (-'(-u))).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
' q - ' q - ' (- u) < y
        rewrite plus_negate_r.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
0 - ' (- u) < y
        rewrite plus_0_l.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
- ' (- u) < y
        rewrite (preserves_negate u).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
- - ' u < y
        rewrite negate_involutive.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
' u < y
        assumption.
      }
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
      assert (sltseps : s<s+' epsilon)
        by apply ltQposQ.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
      destruct (l s (s+' epsilon) sltseps) as [ltsx|ltxseps].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltsx:' s < x
((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltsx:' s < x
((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type apply inl.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltsx:' s < x
' q < x + y apply char_plus_left.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltsx:' s < x
hexists
  (fun s : Q => ((' s < x) * (' (q - s) < y))%type) apply tr; exists s; split; try assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltsx:' s < x
' (q - s) < y
        rewrite preserves_minus; assumption.
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type apply inr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
x + y < ' r apply char_plus_right.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
hexists
  (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type) apply tr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
{t : Q & ((x < ' t) * (y < ' (r - t)))%type}
        set (t := s + ' epsilon); exists t.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
((x < ' t) * (y < ' (r - t)))%type
        split; try assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
y < ' (r - t)
        assert (r-(q-u+(r-q)/2)=u+'epsilon) as ->.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
r - (q - u + (r - q) / 2) = u + ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
y < ' (u + ' epsilon)
        {
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
r - (q - u + (r - q) / 2) = u + ' epsilon
          change ((r - q) / 2) with ('epsilon).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
r - (q - u + ' epsilon) = u + ' epsilon
          rewrite negate_plus_distr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
r + (- (q - u) - ' epsilon) = u + ' epsilon
          rewrite <- negate_swap_l.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
r + (- q + u - ' epsilon) = u + ' epsilon
          rewrite (plus_comm (-q) u).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
r + (u - q - ' epsilon) = u + ' epsilon
          rewrite (plus_assoc r (u-q) (-'epsilon)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
r + (u - q) - ' epsilon = u + ' epsilon
          rewrite (plus_assoc r u (-q)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
r + u - q - ' epsilon = u + ' epsilon
          rewrite (plus_comm r u).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
u + r - q - ' epsilon = u + ' epsilon
          rewrite <- (plus_assoc u r (-q)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
u + (r - q) - ' epsilon = u + ' epsilon
          rewrite <- (plus_assoc u (r-q) (-'epsilon)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
u + (r - q - ' epsilon) = u + ' epsilon
          rewrite (plus_comm r (-q)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
u + (- q + r - ' epsilon) = u + ' epsilon
          rewrite <- (plus_assoc (-q) r (-'epsilon)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
u + (- q + (r - ' epsilon)) = u + ' epsilon
          rewrite path_avg_split_diff_r.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
u + (- q + (r + q) / 2) = u + ' epsilon
          rewrite <- path_avg_split_diff_l.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
u + (- q + (q + (r - q) / 2)) = u + ' epsilon
          rewrite (plus_assoc (-q) q ((r-q)/2)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
u + (- q + q + (r - q) / 2) = u + ' epsilon
          rewrite (plus_negate_l q).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
u + (0 + (r - q) / 2) = u + ' epsilon
          rewrite (plus_0_l _).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
u + (r - q) / 2 = u + ' epsilon
          reflexivity.
        }
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
y < ' (u + ' epsilon)
        assumption.
    Qed.

    (* TODO construct locators for multiplications. *)
    Lemma locator_times : locator (x * y).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
locator (x * y)
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
locator (x * y) Abort.

    Lemma locator_meet : locator (meet x y).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
locator (x ⊓ y)
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
locator (x ⊓ y)
      intros q r ltqr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
((' q < x ⊓ y) + (x ⊓ y < ' r))%type
      destruct (l q r ltqr, m q r ltqr) as [[ltqx|ltxr] [ltqy|ltyr]].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltqx:' q < x
ltqy:' q < y
((' q < x ⊓ y) + (x ⊓ y < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltqx:' q < x
ltyr:y < ' r
((' q < x ⊓ y) + (x ⊓ y < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltxr:x < ' r
ltqy:' q < y
((' q < x ⊓ y) + (x ⊓ y < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltxr:x < ' r
ltyr:y < ' r
((' q < x ⊓ y) + (x ⊓ y < ' r))%type
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltqx:' q < x
ltqy:' q < y
((' q < x ⊓ y) + (x ⊓ y < ' r))%type apply inl, meet_lt_l; assumption.
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltqx:' q < x
ltyr:y < ' r
((' q < x ⊓ y) + (x ⊓ y < ' r))%type apply inr, meet_lt_r_r; assumption.
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltxr:x < ' r
ltqy:' q < y
((' q < x ⊓ y) + (x ⊓ y < ' r))%type apply inr, meet_lt_r_l; assumption.
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltxr:x < ' r
ltyr:y < ' r
((' q < x ⊓ y) + (x ⊓ y < ' r))%type apply inr, meet_lt_r_r; assumption.
    Qed.

    Lemma locator_join : locator (join x y).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
locator (x ⊔ y)
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
locator (x ⊔ y)
      intros q r ltqr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
((' q < x ⊔ y) + (x ⊔ y < ' r))%type
      destruct (l q r ltqr, m q r ltqr) as [[ltqx|ltxr] [ltqy|ltyr]].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltqx:' q < x
ltqy:' q < y
((' q < x ⊔ y) + (x ⊔ y < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltqx:' q < x
ltyr:y < ' r
((' q < x ⊔ y) + (x ⊔ y < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltxr:x < ' r
ltqy:' q < y
((' q < x ⊔ y) + (x ⊔ y < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltxr:x < ' r
ltyr:y < ' r
((' q < x ⊔ y) + (x ⊔ y < ' r))%type
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltqx:' q < x
ltqy:' q < y
((' q < x ⊔ y) + (x ⊔ y < ' r))%type apply inl, join_lt_l_l; assumption.
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltqx:' q < x
ltyr:y < ' r
((' q < x ⊔ y) + (x ⊔ y < ' r))%type apply inl, join_lt_l_l; assumption.
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltxr:x < ' r
ltqy:' q < y
((' q < x ⊔ y) + (x ⊔ y < ' r))%type apply inl, join_lt_l_r; assumption.
      -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F),
' q < x + y <->
hexists
  (fun s : Q =>
   ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F),
x + y < ' r <->
hexists
  (fun t : Q =>
   ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltxr:x < ' r
ltyr:y < ' r
((' q < x ⊔ y) + (x ⊔ y < ' r))%type apply inr, join_lt_r; assumption.
    Qed.

  End binary_ops.

  Section limit.

    Context {xs : nat -> F}.
    Context {M} {M_ismod : CauchyModulus Q F xs M}.
    Context (ls : forall n, locator (xs n)).

    Lemma locator_limit {l} : IsLimit _ _ xs l -> locator l.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
IsLimit Q F xs l -> locator l
    Proof.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
IsLimit Q F xs l -> locator l
      intros islim.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
locator l
      intros q r ltqr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
((' q < l) + (l < ' r))%type
      set (epsilon := (Qpos_diff q r ltqr) / 3).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
((' q < l) + (l < ' r))%type
      (* TODO we are doing trisection so we have the inequality: *)
      assert (ltqepsreps : q + ' epsilon < r - ' epsilon).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
q + ' epsilon < r - ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
((' q < l) + (l < ' r))%type
      {
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
q + ' epsilon < r - ' epsilon
        apply (strictly_order_reflecting (+'epsilon)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
q + ' epsilon + ' epsilon < r - ' epsilon + ' epsilon
        rewrite <- (plus_assoc r (-'epsilon) ('epsilon)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
q + ' epsilon + ' epsilon <
r + (- ' epsilon + ' epsilon)
        rewrite plus_negate_l.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
q + ' epsilon + ' epsilon < r + 0
        rewrite plus_0_r.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
q + ' epsilon + ' epsilon < r
        rewrite <- (plus_assoc q ('epsilon) ('epsilon)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
q + (' epsilon + ' epsilon) < r
        apply (strictly_order_reflecting ((-q)+)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
- q + (q + (' epsilon + ' epsilon)) < - q + r
        rewrite (plus_assoc (-q) q _).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
- q + q + (' epsilon + ' epsilon) < - q + r
        rewrite plus_negate_l, plus_0_l.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
' epsilon + ' epsilon < - q + r
        rewrite (plus_comm (-q) r).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
' epsilon + ' epsilon < r - q
        rewrite <- (mult_1_r ('epsilon)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
' epsilon * 1 + ' epsilon * 1 < r - q
        rewrite <- plus_mult_distr_l.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
' epsilon * 2 < r - q
        unfold epsilon, cast, Qpos_diff; cbn.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
(r - q) / 3 * 2 < r - q
        rewrite <- (mult_assoc (r-q) (/3) 2).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
(r - q) * (/ 3 * 2) < r - q
        pattern (r-q) at 2.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
(fun q0 : Q => (r - q) * (/ 3 * 2) < q0) (r - q)
        rewrite <- (mult_1_r (r-q)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
(r - q) * (/ 3 * 2) < (r - q) * 1
        assert (rqpos : 0 < r-q) by apply (Qpos_diff q r ltqr).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
rqpos:0 < r - q
(r - q) * (/ 3 * 2) < (r - q) * 1
        apply (strictly_order_preserving ((r-q)*.)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
rqpos:0 < r - q
/ 3 * 2 < 1
        apply (strictly_order_reflecting (3*.)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
rqpos:0 < r - q
3 * (/ 3 * 2) < 3 * 1
        rewrite (mult_assoc 3 (/3) 2).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
rqpos:0 < r - q
3 / 3 * 2 < 3 * 1
        rewrite (dec_recip_inverse 3).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
rqpos:0 < r - q
1 * 2 < 3 * 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
rqpos:0 < r - q
3 <> 0
        -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
rqpos:0 < r - q
1 * 2 < 3 * 1 rewrite mult_1_r, mult_1_l.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
rqpos:0 < r - q
2 < 3 exact lt_2_3.
        -
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
rqpos:0 < r - q
3 <> 0 apply apart_ne, positive_apart_zero, lt_0_3.
      }
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
((' q < l) + (l < ' r))%type
      destruct (ls (M (epsilon / 2)) (q + ' epsilon) (r - ' epsilon) ltqepsreps)
        as [ltqepsxs|ltxsreps].
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltqepsxs:' (q + ' epsilon) < xs (M (epsilon / 2))
((' q < l) + (l < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltxsreps:xs (M (epsilon / 2)) < ' (r - ' epsilon)
((' q < l) + (l < ' r))%type
      +
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltqepsxs:' (q + ' epsilon) < xs (M (epsilon / 2))
((' q < l) + (l < ' r))%type apply inl.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltqepsxs:' (q + ' epsilon) < xs (M (epsilon / 2))
' q < l
        rewrite preserves_plus in ltqepsxs.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltqepsxs:' q + ' (' epsilon) < xs (M (epsilon / 2))
' q < l
        assert (ltqxseps : ' q < xs (M (epsilon / 2)) - ' (' epsilon))
          by (apply flip_lt_minus_r; assumption).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltqepsxs:' q + ' (' epsilon) < xs (M (epsilon / 2))
ltqxseps:' q < xs (M (epsilon / 2)) - ' (' epsilon)
' q < l
        refine (transitivity ltqxseps _).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltqepsxs:' q + ' (' epsilon) < xs (M (epsilon / 2))
ltqxseps:' q < xs (M (epsilon / 2)) - ' (' epsilon)
xs (M (epsilon / 2)) - ' (' epsilon) < l
        apply (modulus_close_limit _ _ _ _ _).
      +
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltxsreps:xs (M (epsilon / 2)) < ' (r - ' epsilon)
((' q < l) + (l < ' r))%type apply inr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltxsreps:xs (M (epsilon / 2)) < ' (r - ' epsilon)
l < ' r
        rewrite (preserves_plus r (-'epsilon)) in ltxsreps.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltxsreps:xs (M (epsilon / 2)) <
' r + ' (- ' epsilon)
l < ' r
        rewrite (preserves_negate ('epsilon)) in ltxsreps.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltxsreps:xs (M (epsilon / 2)) < ' r - ' (' epsilon)
l < ' r
        assert (ltxsepsr : xs (M (epsilon / 2)) + ' (' epsilon) < ' r)
          by (apply flip_lt_minus_r; assumption).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltxsreps:xs (M (epsilon / 2)) < ' r - ' (' epsilon)
ltxsepsr:xs (M (epsilon / 2)) + ' (' epsilon) < ' r
l < ' r
        refine (transitivity _ ltxsepsr).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltxsreps:xs (M (epsilon / 2)) < ' r - ' (' epsilon)
ltxsepsr:xs (M (epsilon / 2)) + ' (' epsilon) < ' r
l < xs (M (epsilon / 2)) + ' (' epsilon)
        apply (modulus_close_limit _ _ _ _ _).
    Qed.

  End limit.

End locator.